Inequalities between the norms of a function and its derivatives

The paper is devoted to the problem of finding the maximum of the norm $||x||_q$ with the constraints $||x||_p=\eta$, $||\dot{x}||_r=\sigma$, $x(0)=a$, $a, \sigma, \eta>0$, for functions $x\in L_p(\mathbb{R}_-)$ with derivatives $\dot{x}\in L_r(\mathbb{R_-})$, $0 < p \leqslant q < \infty$, $r > 1$. The arguments employed are based on the standard machinery of the calculus of variations.